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The search for new
Mathematical axioms
Dr Dilip Raghavan
Department of Mathematics
Introduction
Dr Dilip Raghavan
Academic Profiles:
Dr Raghavan received his Ph.D. at the
University of Wisconsin-Madison in May
2008. His advisors were Prof Kenneth
Kunen and Dr Bart Kastermans. He was a
postdoctoral fellow at the University of
Toronto from July 2008 to July 2011. In
2011, he was awarded a JSPS postdoctoral
fellowship and was a JSPS fellow at Kobe
University from July 2011 to August 2012.
He joined the Department of Mathematics,
NUS in August 2012.
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Research Interests:
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•
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Mathematical Logic
Set Theory
General Topology
Contacts Details:
S17-07-15
Department of Mathematics
National University of Singapore
Block S17, 10, Lower Kent Ridge Road
Singapore 119076
Faculty Research Newsletter
Telephone:(+65) 6516-1174
Email
:[email protected]
Webpage :http://www.math.toronto.edu/
raghavan
athematicians prove theorems.
But what does it mean to prove
a theorem? The way it is commonly
understood in Mathematics, it means
to deduce a proposition expressed in a
certain language from a set of axioms
using a fixed collection of inference rules.
This has been the ideal of Mathematics
at least since the time of Euclid, though
it was not always expressed in these
terms. This formulation immediately
raises the issue of how a proposition is
recognized as an axiom. Traditionally
axioms have been thought of as
propositions that "neither need nor
admit of proof". However, perceptions
about which propositions need proof
have changed through history. To take
a famous example, Euclid considered
his fifth postulate to be an axiom. In
two dimensional geometry the fifth
postulate says that given a line L and
point p not on L, there exists exactly
one line through p that does not
intersect L. But mathematicians after
Euclid thought that this needed proof,
and many attempts were made to
prove this from Euclid’s other axioms.
With the recognition of non-Euclidean
geometries in the nineteenth century,
it became clear that the fifth postulate
was indeed independent of Euclid’s
other axioms; that is, it cannot be
proved from his other axioms.
Poincaré Disk
It is instructive to examine in some detail
how the existence of non-Euclidean
geometries shows the independence of
the fifth postulate. One of the simplest
models of a geometry where all of
Euclid’s postulates except the fifth one
hold is known as the Poincaré Disk. Our
"space", the Poincaré Disk, consists of
all points that lie in the interior of some
fixed circle C – i.e. we omit the points
that lie on the circle and take only the
ones inside it. Given points P and Q in
our space, the "line through P and Q"
is interpreted as the arc of the circle
through P and Q that intersects C at
right angles (see Figure 1). With these
interpretations of the words "point"
and "line", it can be shown that the fifth
postulate fails because given a line L
and point p outside L, there will be more
than one line passing through p that
does not intersect L, while Euclid’s other
axioms all remain valid.
Figure 1: The Poincaré disk consists
of the Points in the interior of the
circle C. The "line through P and
Q" and the "line through A and B"
are shown. Note that the Points on
the circle C represent infinity, so
neither "line" has any endpoints.
At this point, you may object that when
Euclid set down his axioms he had in
mind a certain meaning for the words
"point" and "line", and this is not what he
meant. To be sure, Euclid did not have
the Poincaré Disk in mind. However,
the words "point" and "line" occur as
primitive undefined terms in Euclid’s
axioms, and if all you have told me about
points and lines are Euclid’s first four
axioms, then I am free to interpret these
terms any way I want, as long as the four
things you have told me about them
come out true under my interpretation.
Or to put it another way, if all you know about points and lines
are the first four axioms, then you do not know enough to be
able to distinguish between the usual "flat" two dimensional
space and the Poincaré Disk, and the existence of these two
interpretations shows that the fifth postulate is independent
of the other four because it holds in the usual interpretation
while failing in the alternative one1.
formalizing mathematical proofs. The second closely related
trend was the study of arbitrary collections or sets of objects
(usually mathematical objects such as real numbers), without
any fundamental distinction being made between finite and
infinite collections. This was initiated by Cantor, who showed
that there were different kinds of infinite collections – some
infinities were much bigger than others.
The above discussion raises an important point about the
language in which mathematical propositions are expressed.
This language has varied from time to time. For Euclid this
language was Greek. No matter what this language is, it always
contains some primitive undefined terms. The mathematician
who uses these terms may have a certain picture of what they
represent in his or her mind, but while proving a theorem only
the information that is declared in the axioms may be used.
The language of modern mathematics is an extremely precise
and wholly artificial language in which only a small number of
primitive terms occur. In fact, apart from some logical terms
connected with reasoning in general, only one primitive term
occurs: the notion of a set2. The rules for forming a meaningful
sentence in this language are so precise, the axioms governing
the usage of the primitive terms are so easy to list, and the
rules of inference sufficiently well-defined, that it possible for
a computer program to recognize when a certain string of
symbols is a meaningful sentence in this language, and when
a sequence of such sentences constitutes a proof. Needless
to say, mathematicians do not think in terms of this artificial
language, but all theorems and proofs can, at least in principle,
be translated into it.
This trend towards studying arbitrary functions and sets
eventually led to some paradoxes that resulted when certain
sets with self-referential definitions were considered. To give a
simple illustration of these paradoxes, suppose that there is a
town in which there is a barber, who is male, and he shaves all
and only those men in the town who do not shave themselves.
Does the barber shave himself? By our hypothesis he shaves
himself if and only if he does not. So such a barber simply
cannot exist. Similarly it was realized that sets with such selfreferential definitions cannot exist either, and axioms specifying
exactly what kinds of sets exist were drawn up. Thus was born
the axiom system known as ZFC, which is expressed in the
artificial language referred to above. The axioms of ZFC provide
all the information about sets that is relevant to mathematics,
including information about what sets exist. All theorems
proved by mathematicians today can be translated into the
language of ZFC and proved from its axioms3.
Why did modern mathematics evolve such an artificial
language and what is the notion of a set? This is a complex
story, but a very brief outline is as follows. In the nineteenth
century several mathematicians felt a need for greater rigour
in dealing with the concepts of calculus such as the concept of
a limit, which till then were based on geometric intuition. This
led them to redefine these notions in terms of just real numbers
and operations performed on them. Then real numbers were
reduced to the rational numbers, which in turn were reduced
to the familiar natural numbers of arithmetic. This trend went
hand in hand with two other trends. The first was an expansion
in the sense of the term "function". Functions, which had
initially denoted certain mathematical expressions, came to
denote arbitrary correspondences. This led Frege to think of
predicates and relations as functions that map objects to truth
values. For example, he thought of the relation "is the father
of" as a function whose arguments are a pair of objects x and
y and it returns the value "true" if x is the father of y and the
value "false" otherwise. Other functions could then be defined
that took such functions as their arguments, and this process
could be iterated. He built an artificial language using such
functions and showed that this language was very useful for
1.
For more on non-Euclidean geometries see [1]. You may
also wish to google "Escher’s Circle Limit ".
2. Technically, it is the relation of set membership.
But is it the case that the axioms of ZFC can, at least in
principle, be used to settle all mathematical questions? The
answer is no, and in fact this answer is not unique to ZFC.
In the 1930 Gödel proved that there there are propositions
in the language of ZFC that are independent of the axioms
of ZFC. That is, there are sentences in the language of ZFC
that stand in the same relation to its axioms as Euclid’s fifth
postulate stands in relation to his other four – these sentences
can neither be proved nor disproved using the axioms of ZFC.
So for any one of these independent sentences, there exist
two different interpretations of the term "set", one in which
the sentence is true and another one in which the sentence
is false, even though all the axioms of ZFC hold under both
interpretations. All of the techniques of modern mathematics
can be implemented with ZFC; therefore these independent
sentences express mathematical questions that cannot be
resolved using any mathematical technique, unless of course,
one is willing to accept new axioms. It is worth noting that
this predicament is not unique to ZFC or to the concept of
"set". Gödel showed that for any axiom system that is powerful
enough for the development of modern mathematics and
whose axioms are simple to list, there will be sentences in the
language of that axiom system that are independent of it, and
this is regardless of the primitive concepts used4.
A large part of modern set theory is devoted to constructing
various different interpretations of the term "set". Just as the
3. Consult [2] for the original sources relating to the
development of the artificial language in which ZFC is
expressed, for the development of the set concept, and
for the evolution of the axioms of ZFC.
4. Gödel’s original papers are in [2].
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Faculty Research Newsletter
Notion of a Set
ZFC Axiom System
interpretations of the terms "point" and "line" were carefully
constructed in the Poincaré Disk example to ensure that
Euclid’s first four postulates were true, these interpretations
of "set" are carefully constructed to ensure that the axioms
of ZFC are all true. Such an interpretation is known as a
model of ZFC. Now, given a mathematical sentence, if there
is a model of ZFC in which that sentence is true and also a
model where it is false, then the question of whether that
sentence is true cannot be answered by usual mathematical
techniques. One of the aims of investigating models of ZFC
is this kind of delimitation of possibilities. It is worth noting
here that these models are constructed using tools available
in ZFC (this is similar to the Poincaré Disk being defined using
Euclidean circles)5. So this is an exploration of the limits of
mathematics from within itself. Another aim of this study is to
discover new possible axioms. The traditional view of axioms
was that they have to be self-evident. However, we may also
discover sentences independent of ZFC, which when added
as new axioms give such a coherent and appealing picture of
some class of mathematical structures that we are tempted
to enlarge ZFC in that direction. In fact, two distinct types of
such sentences have been recognized over the last several
decades of research. The first class may be called "minimality
hypotheses". Very roughly, they say that the universe of sets
is as thin as possible. The picture they present of certain
infinite combinatorial structures is one of "non—structure",
meaning that they imply that such combinatorial structures
are disparate and impossible to classify. The other class
of sentences are collectively known as "forcing axioms".
Very roughly, they say that the universe of sets is as wide as
possible. They tend to imply that a structure theory exists for
several classes of combinatorial structures, meaning that the
objects in these classes are not so diverse and must belong
to one of a small number of types6. Another aim of studying
models of ZFC is to discover new theorems of ZFC. Sometimes
showing that a statement is independent of ZFC can suggest
candidates for new theorems and can provide hints as to what
techniques to use for proving them. In [4] Shelah calls this the
"Rubble Removal Thesis": after the independent statements
are cleared away, we may be left with unexpected theorems.
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Faculty Research Newsletter
In my own work, I have made small contributions to this ongoing study of models of ZFC. To illustrate, in [5] a limit on
possibilities is set by showing that the existence of a certain
set of real numbers is independent of ZFC. In [6], I have studied
the consequences of some forcing axioms. In [7], I prove a
theorem of ZFC, which could have been discovered classically,
but was found only after several independence results in its
vicinity were proved, and its proof uses techniques that are
reminiscent of independence proofs.
5. There is a technical issue that is being swept under the
rug here. Strictly speaking, it is not possible to construct
a model of ZFC within ZFC, but only models of arbitrarily
large finite fragments of it. This turns out to be good
enough.
6. For more on forcing axioms and structure theorems for
classes of combinatorial structures, see [3].
References
1. Coxeter, H. S. M., "Non—Euclidean geometry",
Mathematical Association of America: Washington, DC
(1998).
2. Van Heijenoort, J. (ed.), "From Frege to Gödel", Harvard
University Press: Cambridge, MA (2002).
3. Moore, J. T., "The proper forcing axiom", Proceedings of the
International Congress of Mathematicians., Vol II, pp. 3—29
(2010).
4. Shelah S., "Logical dreams", Bull. Amer. Math. Soc. (N.S.),
Vol 40, pp. 203—228 (2003).
5.
Raghavan D., "A model with no strongly separable almost
disjoint families", Israel J. Math., Vol 189, pp. 39—53
(2012).
6. Raghavan D., "P-ideal dichotomy and weak squares", J.
Symbolic Logic, to appear.
7.
Raghavan D., "There is a Van Douwen mad family", Trans.
Amer. Math. Soc., Vol 362, pp. 5879—5891 (2010).